Stochastic Processes at CEU -- winter semester 2018/19

HOMEWORKS AND OTHER EXERCISES (will be)here.
Homeworks to hand in by the 19th of February: 1.3; 1.4; 1.7; 1.9; 1.10

No. of Credits: 3
No. of ECTS credits: 6
Time Period of the course: Winter Semester
Prerequisites: Probability 1
Course Level: PhD
Syllabus: will be here.
Classes: Tuesdays from 09:00 in room 301.
Course Coordinator: Imre Péter Tóth.

Planned schedule

week #	when	topic	remark
week 1	2019.01.08	Introduction, overview. Finite dimensional distributions and construction of stochastic processes. Kolmogorov extension theorem.
week 2	2019.01.15	Classification of stochastic processes, examples. Poisson process, Wiener process. Markov property, finite Markov chains.
week 3	2019.01.22	Mixing and ergodicity of Markov chains. Countable Markov chains. Transience, recurrence, positive recurrence. Random walks on Z^d, Pólya's theorem.
week 4	2019.01.29	Random walks on Z. The reflection principle and applications. Distribution of the maximum, arcsine law, return times, local time.	CLASS DELAYED to the next three weeks
week 5	2019.02.05	Discrete time martingales. Branching processes. Barabási-Albert graph model, preferential attachment.	CLASS LONGER with 35 minutes + a break
week 6	2019.02.12	Discrete time stochastic integration. Discrete Black-Scholes formula.	CLASS LONGER with 35 minutes + a break
week 7	2019.02.19	Brownian motion, Wiener process. Construction(s), regularity properties, scaling properties.	CLASS LONGER with 35 minutes + a break
week 8	2019.02.26	Markov chains in continuous time. Infinitesimal generator, exponential semigroup. Examples.
week 9	2019.03.05	Filtrations and stopping times. Markov processes and martingales in continuous time.
week 10	2019.03.12	Ito's stochastic integral. Motivation, definition, basic properties.
week 11	2019.03.19	Itô processes and the Itô formula. Examples of stochastic differential equations.
week 12	2019.03.26	Itô representation theorem, Martingale representation theorem.

Suggested literature:

• R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
• R. Durrett: Essentials of Stochastic Processes. 2nd edition, Springer, 2012.
• S: Karlin, H.M. Taylor: A First Course in Stochastic Processes. 2nd edition, Academic Press 1975.
• S: Karlin, H.M. Taylor: A Second Course in Stochastic Processes. Academic Press 1981.
• R. Durrett: Stochastic Calculus: A practical Introduction. CRC Press, 1996.
• F.C. Klebaner: Introduction to stochastic calculus with applications. 2nd edition, Imperial College Press, 2005.
• K.L. Chung, R.J. Williams: Introduction to stochastic integration. 2nd edition, Birkhauser 1990.
• D. Revuz, M. Yor: Continuous Martingales and Brownian Motion. 3rd edition, Springer 1999.

Grading rules:
There will be three homework assignments -- roughly once in every four weeks -- worth 20% of the total score. A midterm exam focused on problem solving will be worth 30%, while the final exam, focused on theory will be worth 50%.

Requirements for "audit":
regular participation in class and a short oral account of the concepts and phenomena learned.